Much time and effort has been expended in recent decades to develop better volatility models that can improve volatility forecasts, which many financial applications typically use to forecast future returns and volatility associated with securities, bonds, commodities, and other financial instruments. In particular, financial applications that tend to use volatility forecasts include managing risk, pricing and hedging derivatives, timing markets, and constructing portfolios, among others. In these and other related contexts, the predictability associated with the volatility model used to make forecasts tends to be an important factor in determining whether the forecasts can be considered reliable. For example, suitably managing financial risk often requires present knowledge relating to the likelihood that a portfolio will have a future decline in value or whether certain stocks or portfolios should be sold prior to becoming excessively volatile. In another example, developing strategies to effectively trade option contracts may require insight into the volatility that can be expected over a lifetime associated with the option contract, while market makers may widen the bid-ask spread associated with option contracts expected to have high future volatility. Further background relating to volatility models and various challenges associated therewith may be provided in “What Good is a Volatility Model?” and “Modeling and Forecasting Realized Volatility,” the contents of which are hereby incorporated by reference in their entirety.
Among single factor stochastic volatility models, the Heston single factor model tends to be the most popular and easiest to implement. In the Heston single factor model, the volatility follows an Ornstein-Uhlenbeck process that generally describes how certain variables linearly transform towards a long-term mean over time, a common economic assumption that applies to stock prices and stock options. As such, to solve the associated pricing and hedging problem, the Heston volatility model may perform a Fourier transform on the probability distribution function associated with the Ornstein-Uhlenbeck process. Later research extended the Heston model to a multi-factor volatility process that can realize long-term and short-term volatilities in addition to implied and skew volatilities, while related research derived option pricing schemes using mathematics that attempted to explain the “smile,” skew, and other stylized volatility facts. The connection between the Heston model, described in “A Closed Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” and the later research relating to multi-factor models can be found in the underlying mathematics that invert a function that helps solve the volatility problem, as described in “A Multifactor Volatility Heston Model” and “Dynamics of Markets,” all of which are hereby incorporated by reference in their entirety.
Building on the Heston single factor model and the related multi-factor models, a multi-fractal model where the fractal value or Hausdorff dimension varies over time were applied to the volatility context. In particular, the multi-fractal model generally assumes that shocks to volatility will be scale invariant, in that daily or short-term shocks will have a long-term volatility impact in much the way that short-term volatility will be impacted. The multi-fractal model, described in “Multifractal Volatility,” the contents of which are hereby incorporated by reference in their entirety, was claimed to capture the stylized facts associated with volatility, including long memory and intermittency, more effectively than the Heston model and the multi-factor models. However, these claims rest on misleading assumptions common to the multi-fractal interpretation and many standard economic and finance models. For example, standard economic and financial theory provides that market makers instantaneously integrate new price information, which leads to little (if any) attention to the lag between the time when new price information becomes available and the time when the market makers learn about the new price information. Moreover, different market makers may interpret the same news differently, and movements in volatility are not the same at all time scales. Accordingly, the prevailing volatility models, while capturing many important stylized facts associated with volatility, tend to fall short in providing the requisite predictability and stability because certain factors may be marginalized or unconsidered.
In recent years, increases in data availability over shorter time horizons has prompted efforts to forecast volatility over weekly, daily, and other high-frequency time periods and substantial growth in markets associated with volatility swaps, variance swaps, and other derivatives that incorporate information forecasted with volatility models. For example, a volatility swap generally relates to a forward contract on the future realized volatility associated with a particular underlying asset, whereby volatility swaps allow investors to directly trade on the volatility associated with the underlying asset in a substantially similar manner to how investors typically trade on a price index. As such, volatility swaps provide financial instruments that can be used to speculate on future volatility levels or hedge volatility exposure associated with other positions or businesses. In contrast, a variance swap generally relates to a financial derivative that allows investors to trade on the realized volatility associated with a stock index, interest rate, exchange rate, or similar underlying assets (e.g., the VIX ticker symbol represents the implied volatility associated with S&P 500 index options over a next thirty-day period). In this context, the main difference between a volatility swap and a variance swap relates to the profit and loss from a variance swap depending directly on the difference between realized and implied volatility, whereas' a volatility swap depends only on the realized volatility. Despite variance swaps depending on sometimes uncertain and unpredictable implied volatility metrics, commentators discussing the option markets typically use the VIX implied (or expected) volatility measure to represent the overall market sentiment associated with equity options.
However, the relationship between the VIX and individual equity options tends to be overstated, in that the dynamics that drive the volatility associated with index options are different from the dynamics that drive the volatility associated with equity options, and the volatility associated with the two may not correlate in many cases. In particular, VIX quotes are expressed in percentage points, and as noted above, roughly translate to the expected movement in the S&P 500 index over the next thirty-day period on an annualized basis. For example, if the VIX has a current quote at fifteen, the quote represents an expected fifteen percent annualized change over the next thirty days. From this expected annualized change, an analyst can infer that the index option markets expect the S&P 500 to move up or down approximately 4.33 percent over the next thirty days (i.e., fifteen percent divided by the square root of twelve). Thus, S&P 500 index options would be priced with the assumption that a sixty-eight percent (one standard deviation) likelihood exists that the return on the S&P 500 over the next thirty days will have a 4.33 percent magnitude (up or down). Consequently, the VIX does not necessarily provide a suitable measure relating to the market sentiment associated with individual equity options because the VIX measures volatility over a thirty-day period, while the most liquidity associated with non-index equity options tends to be found in the two to six-month maturities. Further, the volatility associated with individual equities typically depends on market sector (e.g., technology stocks are typically assumed to have high volatility, while utility stocks are typically assumed to have low volatility). Accordingly, using one number (e.g., the VIX) to represent the volatility associated with all equity options tends to be overly simplistic, and moreover, carries uncertainty and unpredictability because the VIX and other variance swaps are coincident volatility indicators rather than forward volatility indicators.